Zero Complement is Not Empty

Theorem

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

Let $S^*$ be the zero complement of $S$.

Then $S^*$ is not empty.

Proof

From Naturally Ordered Semigroup Axiom $\text {NO} 4$: Existence of Distinct Elements, we have:

$\exists m, n \in S: m \ne n$

That is, there are at least two distinct elements in $S$.

Therefore, there must be at least one element in $S^* = S \setminus \set 0$.

So:

$S^* = S \setminus \set 0 \ne \O$

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers